Global solutions to the Chemotaxis-Navier-Stokes equations with some large initial data

In this paper, we mainly study the Cauchy problem of the Chemo-taxis-Navier-Stokes equations with initial data in critical Besov spaces. We first get the local wellposedness of the system in $\mathbb{R}^d \, (d≥2)$ by the Picard theorem, and then extend the local solutions to be global under the only smallness assumptions on $\|u_0^h\|_{\dot{B}_{p, 1}^{-1+\frac{d}{p}}}$, $\|n_0\|_{\dot{B}_{q, 1}^{-2+\frac{d}{q}}}$ and $\|c_0\|_{\dot{B}_{r, 1}^{\frac{d}{r}}}$. This obtained result implies the global wellposedness of the equations with large initial vertical velocity component. Moreover, by fully using the global wellposedness of the classical 2D Navier-Stokes equations and the weighted Chemin-Lerner space, we can also extend the obtained local solutions to be global in $\mathbb{R}^2$ provided the initial cell density $n_0$ and the initial chemical concentration $c_0$ are doubly exponential small compared with the initial velocity field $u_0$.